under the continuity equation constraint (REF ) and the time-boundary constraints \(\mu (t=0,\cdot ) = \alpha \)  and \(\mu (t=1,\cdot ) = \beta \) . Here, we consider
a function \(f\)  such that \(f(x,\cdot ,\cdot ,\cdot )\)  is nonnegative, convex, positively \(1-\) homogeneousWe refer the reader to [1]} for further details on the function \(f\) .. In comparison with the Lagrangian in Equation (REF ), we incorporate \(\mu \)  in the Lagrangian and we impose its \(1-\) homogeneity. In addition, \(\lambda (t,x)\)  stands for a measure in time and space dominating all other measures \(\mu ,\omega \)  and \(\xi \)  and due to the hypothesis on \(f\) , Definition (REF ) does not depend on the choice of \(\lambda \) .
